Tensor-on-Tensor Regression

Lock, Eric F.

doi:10.1080/10618600.2017.1401544

Cited by 87 publications

(92 citation statements)

References 52 publications

(69 reference statements)

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“…More general problems, where the unknowns are matrices, for example,

\min_{X, Y} ‖ A \cdot {(X, Y)}_{2, 3} - B ‖,

are studied in the works of Hoff and Lock . The methods of this paper can be adapted to such problems.…”

Section: Discussionmentioning

confidence: 99%

“…Assume that the perturbations  and b satisfy (12). Consider first the perturbation of the residual, r + r = ( + ) · (x + x, + ) 2,3 − b − b = r +  · ( x, ) 2,3 +  · (x, ) 2,3 +  · (x, ) 2,3 − b + O( 2 ).…”

Section: Appendix a : Derivation Of The Perturbation Equation 14mentioning

confidence: 99%

“…The problem can also be considered as a bilinear tensor regression problem . In the last couple of years, such problems are becoming quite abundant in the literature; see, for example, two recent papers dealing with more general setups . However, most such papers are application oriented and do not deal with basic properties and algorithms (in fact, most propose alternating algorithms).…”

Section: Introductionmentioning

confidence: 99%

“…In the last couple of years, such problems are becoming quite abundant in the literature; see, for example, two recent papers dealing with more general setups. 9,12 However, most such papers are application oriented and do not deal with basic properties and algorithms (in fact, most propose alternating algorithms). In this paper, we are concerned with the simpler problem (1), we introduce perturbation analysis, and we discuss and implement algorithms for problems of small and medium dimensions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solving bilinear tensor least squares problems and application to Hammerstein identification

Eldén

Ahmadi‐Asl

2018

Numerical Linear Algebra App

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Summary Bilinear tensor least squares problems occur in applications such as Hammerstein system identification and social network analysis. A linearly constrained problem of medium size is considered, and nonlinear least squares solvers of Gauss–Newton‐type are applied to numerically solve it. The problem is separable, and the variable projection method can be used. Perturbation theory is presented and used to motivate the choice of constraint. Numerical experiments with Hammerstein models and random tensors are performed, comparing the different methods and showing that a variable projection method performs best.

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“…More general problems, where the unknowns are matrices, for example,

\min_{X, Y} ‖ A \cdot {(X, Y)}_{2, 3} - B ‖,

are studied in the works of Hoff and Lock . The methods of this paper can be adapted to such problems.…”

Section: Discussionmentioning

confidence: 99%

Section: Appendix a : Derivation Of The Perturbation Equation 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving bilinear tensor least squares problems and application to Hammerstein identification

Eldén

Ahmadi‐Asl

2018

Numerical Linear Algebra App

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“…Many researchers have studied matrix-and tensor-variate regression (Zhou et al 2013, Zhou & Li 2014, Zhao & Leng 2014, Hoff 2015, Raskutti & Yuan 2015, Sun et al 2016, Wang & Zhu 2016, Lock 2017. But most of these methods do not directly apply to classification or incorporating the covariates.…”

Section: Introductionmentioning

confidence: 99%

Covariate-Adjusted Tensor Classification in High Dimensions

Pan

Mai

Zhang

2018

Journal of the American Statistical Association

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In contemporary scientific research, it is of great interest to predict a categorical response based on a high-dimensional tensor (i.e. multi-dimensional array) and additional covariates. This mixture of different types of data leads to challenges in statistical analysis. Motivated by applications in science and engineering, we propose a comprehensive and interpretable discriminant analysis model, called CATCH model (in short for Covariate-Adjusted Tensor Classification in High-dimensions), which efficiently integrates the covariates and the tensor to predict the categorical outcome. The CATCH model jointly models the relationships among the covariates, the tensor predictor, and the categorical response. More importantly, it preserves and utilizes the structures of the data for maximum interpretability and optimal prediction. To tackle the new computational and statistical challenges arising from the intimidating tensor dimensions, we propose a penalized approach to select a subset of tensor predictor entries that has direct discriminative effect after adjusting for covariates. We further develop an efficient algorithm that takes advantage of the tensor structure. Theoretical results confirm that our method achieves variable selection consistency and optimal classification error, even when the tensor dimension is much larger than the sample size. The superior performance of our method over existing methods is demonstrated in extensive simulated and real data examples.

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Bayesian Methods for Tensor Regression

Guhaniyogi¹

2020

Wiley StatsRef: Statistics Reference Online

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For many applications pertaining to neuroimaging, social science, international relations, chemometrics, genomics, and molecular omics, datasets often involve variables which are best represented in the form of a multidimensional array or tensor , which extends the familiar two‐way data matrix into higher dimensions. Rather than vectorizing tensor‐valued variables prior to analysis which results in loss of inference, new methods have emerged developing regression relationships between variables with either tensor‐valued response(s) or predictor(s). Bayesian approaches, in particular, have shown great promise in applications pertaining to tensor regressions. A remarkable feature of fully Bayesian approaches is that they allow flexible modeling of tensor‐valued parameters in the regressions involving tensor variables and naturally offer characterization of uncertainty in the parametric and predictive inferences. This article provides a review of some relevant Bayesian models on tensor regressions developed in recent years. We divide methods according to the objective of the analysis. We begin with tensor regression approaches with a scalar response and a tensor‐valued covariate and discuss both parametric and nonparametric modeling options and applications in this framework. We then address the problem of making inference with a tensor response and a vector of covariates, with applications including task‐related brain activation and connectivity studies. Finally, we offer discussion on Bayesian models involving a tensor response and a tensor covariate. Discussion of each model is accompanied by available results on its posterior contraction properties, laying out restrictions on key model parameters (such as the tensor dimensions) to draw accurate posterior inference.

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Tensor-on-Tensor Regression

Cited by 87 publications

References 52 publications

Solving bilinear tensor least squares problems and application to Hammerstein identification

Solving bilinear tensor least squares problems and application to Hammerstein identification

Covariate-Adjusted Tensor Classification in High Dimensions

Bayesian Methods for Tensor Regression

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